Questions tagged [d-modules]
Modules over rings of differential operators.
265
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Riemann-Hilbert problem via quiver description
The moduli space of Fuchsian systems over $\mathbb{P}^1$ with prescribed adjoint orbits conditions at poles a.k.a. additive Deligne-Simpson problem can be presented under purely quiver description.The ...
5
votes
1
answer
301
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Operations on perverse sheaves on disk
The category of perverse sheaves on the disk is isomorphic to the category of diagrams
$$E\substack{\substack{c\\\to}\\\substack{v\\\leftarrow}}F$$
With $E,V$ finite dimensional vector spaces, and ...
4
votes
1
answer
191
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Are perverse sheaves representations of some topological invariant?
The well-known correspondence between vector bundles with flat connection on a smooth complex algebraic variety $X$ and complex representations of $\pi_1(X^{an})$, the fundamental group of the ...
3
votes
1
answer
364
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F-crystals from crystalline cohomology
In Section 7 of Katz' paper:
https://web.math.princeton.edu/~nmk/old/travdwork.pdf
He asserts "Crystalline cohomology tells us that for each integer $i \geq 0$, the de Rham cohomology $H^i=Rf_*(\...
2
votes
0
answers
112
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Lie Algebra representations outside of generalized central characters
For a simple Lie algebra $\mathfrak{g}$, we can view its category of representations as fibered over $\operatorname{Spec}Z(\mathfrak{g})$ (a representation will lie over a point if the center's action ...
2
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0
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89
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Applications of the Riemann-Hilbert Correspondence
I am aware of the (first) proof of the Kazhdan–Lusztig conjectures using the Riemann-Hilbert Correspondence. Are there any other interesting applications of the RH correspondence?
2
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0
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305
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Modern treatment of $q$-differential operators/$\mathcal{D}_q$ modules?
The basic idea of $q$-differential operators: replace
$$\partial\cdot x^n\ =\ nx^{n-1} \hspace{10mm}\rightsquigarrow\hspace{10mm} \partial\cdot x^n \ =\ [n]_q x^{n-1} $$
where $[n]_q=(q^{n}-1)/(q-1)$ ...
3
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0
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194
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What is a twisted D-module?
Let $X/\mathbb{C}$ be an abelian variety, $Y$ be the dual abelian variety, and $P$ be the Poincaré bundle on $X\times Y$. On p.207, Correction to “Sheaves with connection on abelian varieties” (by M. ...
2
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0
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125
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Compact generators of $\mathcal{D}\text{-Mod}$ via Mayer Vietoris
Let $X$ be a complex variety (or any space for whom a category of sheaves with the six functors is defined), and
$$i\ :\ Z\ \to\ X\ \leftarrow\ U\ :\ j$$
be complementary open and closed embeddings.
...
9
votes
0
answers
388
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What's a holonomic D-module from the point of view of de Rham spaces?
Let $X$ be a smooth algebraic variety over $\mathbb{C}$. We can consider its de Rham space $X_\text{dR}$ as the sheaf on $(\textsf{Sch}/\mathbb{C})_\text{ét}$ defined by $X_\text{dR}(S):= X(S_\text{...
4
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180
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D-modules generated by derivatives of Delta function
We consider D-modules over the affine line $\mathbb{A}_{\mathbb{C}}^{1}$, i.e. modules of Weyl algebra $A_1(\mathbb{C})=\mathbb{C}[x,\partial]$. There is a set of microfunctions (ref. A Primer of ...
8
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1
answer
277
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What are twisted Verma modules? Basic properties?
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$, $\lambda$ be a weight in the nonnegative Weyl chamber and $w_1,w_2\in W$. Then the twisted Verma modules are defined e.g. in Andersen and ...
4
votes
1
answer
390
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Direct image of Lagrangian subspaces of the co-tangent bundle
Let $p:X \to Y$ be a map of smooth algebraic varieties.
Let $C \subset T^* X$ be a (locally closed) submanifold. Denote by $p_*(C) \subset T^* Y$ the following set:
$$ \{(y,v) \in T^*(Y)\mid\exists ...
6
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257
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Naïve pushforward of D-modules and Gauss–Manin connection
Suppose that $f\colon X\to Y$ is a morphism of smooth quasi-projective varieties over a field of characteristic $0$. We then have a naïve pullback functor $f^\circ:=\mathcal D_{X\to Y}\otimes_{f^{-1}\...
3
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0
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224
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Confusion about definition of crystals
In the notes by Lurie there seems to be two possible definition for crystals which both makes sense for arbitrary functors $X : \mathrm{CRing}_k \to \mathrm{Set}$. ($k$ here is a field. We probably ...
13
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2
answers
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Can we use Mann's six-functor formalism with D-modules?
In a recent course in Bonn, P. Scholze explains a formalization of a six-functor formalism due to L. Mann. In this axiomatization, three of the functors $f_!,f^*,\otimes$ are "constructed" (...
2
votes
1
answer
216
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D modules over nodal curves
Is there a simple description for $D$ modules over $\text{Spec}\left(k\left[x,y \right] / \left(xy\right) \right)$ (say k is algebraically closed of characteristic 0)? Is there a description of the $D$...
3
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0
answers
108
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Local cohomology with coefficients in ideals of parameters
I'm not an expert in local cohomology, but the following problems have come up in my work, and I'd like to get a sense of where things stand.
Let $\mathbb{A}^n=\operatorname{Spec} \mathbb{C}[x_1, \...
4
votes
0
answers
96
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Confusion about D-affineness and jet sheaves on projective line
I have encountered what feels like a very basic confusion relating to the Beilinson Bersnstein localisation theorem.
This theorem in particular states that if $F$ is a $D_{\mathbb{P}^{1}}$ module, ...
15
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2
answers
985
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Examples of Stokes data
I'm trying to learn Stokes data but can't find an example to get my teeth into it.
Background. It's well-known that on a complex manifold $X$, there is the Riemann Hilbert equivalence
$$\text{regular ...
2
votes
0
answers
156
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Smooth pullback of holonomic D-modules is fully faithful
Let $X$ and $Y$ be (not necessarily smooth) algebraic varieties over an algebraically closed field of characteristic $0$, and suppose we have a smooth surjective map $f: X \to Y$ of relative dimension ...
2
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0
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89
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Generalizations of elliptic chain complexes
I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry....
10
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2
answers
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Relation between holonomic D-modules and perverse sheaves
Given a smooth complex algebraic variety, the Riemann-Hilbert-correspondence tells us, that the category of perverse sheaves is equivalent to the category of regular, holonomic D-modules.
However not ...
26
votes
2
answers
2k
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Codimension of the range of certain linear operators
Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map
$D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the differential operator corresponding ...
2
votes
1
answer
285
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Open/closed embeddings and the de Rham space
Let $U\to X$ be an open immersion of schemes and denote by $D$ the (say reduced) complement. Then by applying the de Rham functor, we get morphisms
$$U_{dR}\to X_{dR}\leftarrow D_{dR}$$
of the ...
1
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0
answers
87
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Good D-module is induced by coherent O-module?
Let $X$ be a compact complex manifold. In Definition 4.24, p.78 of D-modules and Microlocal Calculus (by Kashiwara), a coherent $D_X$-module $F$ is called good if there is a directed family $\{G_i\}...
2
votes
1
answer
197
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About the support of a holonomic D-module
Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and let $M^\bullet\in\mathsf{D}^b_\text{h}(\mathcal{D}_X)$ be a complex of D-modules with holonomic cohomologies. We define the support of $M^\...
16
votes
2
answers
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The algebraic version of Riemann-Hilbert correspondence
It is well known that if I have a differentiable manifold (holomorphic manifold) $M$, then I have a functor from the category of vector bundles on $M$ with flat connections to the category of local ...
6
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296
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D-modules on singular varieties; forgetful functors, and t-structures
Let $Z$ be a singular variety over the complex numbers with a closed embedding $i: Z \hookrightarrow X$ into a smooth variety $X$. One can define the derived category $\mathcal{D}(Z)$ of D-modules on $...
3
votes
1
answer
243
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Artin vanishing for D-modules (i.e., when is $f_+$ t-exact?)
Let $f:X\to S$ be a morphism between algebraic varieties which are smooth over a field of characteristic zero. We define the (derived) direct image functor $f_+:\mathsf{D}^b(\mathcal{D}_X)\to \mathsf{...
4
votes
0
answers
90
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Holonomic distributions in the analytic setting
We are interested in a reference to the notion of Holonomic distributions in the real analytic setting. Namely, distributions that generate a Holonomic D-module under the action of the algebra of ...
8
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4
answers
3k
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Gluing perverse sheaves?
It might be a stupid question.
How to glue perverse sheaves? I am considering the following example. Flag variety of $sl_2$, which is $P^1$. Consider category of perverse sheaves on $P^1$. Denoted by $...
8
votes
1
answer
463
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D-modules as ind-coherent sheaves over positive characteristics?
There is an interpretation of D-modules over "sufficiently nice" prestacks $X$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I....
1
vote
1
answer
238
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Every rank 1 local system $L$ satisfying $m^*L=L\boxtimes L$ comes from the Lang torsor? The same holds for D-modules?
Let $G$ be a commutative connected algebraic group over a field $k$, with group operation $m:G\times G\to G$. If $k=\mathbb{F}_q$, we may use a character $\varphi:G(k)\to\overline{\mathbb{Q}}_\ell^\...
20
votes
3
answers
4k
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What is a twisted D-Module intuitively?
When I think about $\mathcal{D}$-Modules, I find it very often useful to envison them as vectorbundles endowed with a rule that decides whether a given section is flat. Or alternatively a notion of ...
5
votes
0
answers
144
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Concrete computation of pullback of the $D$-module $\mathbb{C}[t,t^{-1}]$
I have some problems in calculating some example explicitly. Consider
$$ \{0\} \overset{i}{\rightarrow} \mathbb{C} \overset{j}{\leftarrow} \mathbb{C}^*.$$
Then $Rj_+\mathbb{C}[t,t^{-1}] = \mathbb{C}[t,...
3
votes
1
answer
150
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Explicit computation of D-modules pullback
Consider the $D_{\mathbb{A}^1}$-module $M:=D_{\mathbb{A}^1}/(x)$, and the map $f:z\mapsto z^k$. I want to know $f^*(M)$. I believe it only has a single non-zero cohomology, namely in degree $0$, which ...
83
votes
2
answers
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Why is differential Galois theory not widely used?
E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
5
votes
1
answer
256
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Two identities involving Ext functors in the context of D-modules
I have several questions regarding proposition 2.3 in "Cherednik and Hecke algebras of varieties with a finite group action", by Pavel Etingof. Let $X$ be a complex affine algebraic variety ...
28
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8
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Applications of microlocal analysis?
What examples are there of striking applications of the ideas of microlocal analysis?
Ideally I'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/...
2
votes
1
answer
170
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Confusion about Wakimoto's chiral differential operators on $\mathbf{P}^1$
It is a classical result of Wakimoto that the sheaf of chiral differential operators $D_{ch}$ on $\mathbf{P}^1$ has global sections
$$D_{ch}(\mathbf{P}^1)\ \simeq\ L_{-2}(\mathfrak{sl}_2)$$
the simple ...
5
votes
1
answer
265
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Holonomic = annihilated by some differential operator
Let $X$ be a smooth variety over a characteristic zero field $k$. It seems to be well-known that
"A coherent $\mathcal{D}_X$-module is holonomic if and only if 'every element is annihilated by a ...
16
votes
2
answers
977
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What is the motivation behind the characteristic variety of a D-module and what does it's geometry tell me about the D-module?
Given a smooth algebraic variety $X$, and an $\mathcal{M}\in \text{Mod}(D_X)$, there is the characteristic variety of $\mathcal{M}$ defined as
$$
\text{Char}(\mathcal{M}):= V\left(\sqrt{Ann(\mathcal{M}...
4
votes
0
answers
135
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D-module theoretic Chern characters and an index-type theorem
Let $X$ be a smooth projective variety over $\mathbf{C}$. Consider the category $\mathbf{D}_{X}^{\text{perf}}$ of perfect complexes of left $D$-modules on $X$. It is well known that there is an ...
3
votes
1
answer
323
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Koszul complex of a $\mathcal{D}$-module
I am trying to understand the construction of push forward in the derived category of $\mathcal{D}$-modules using the Koszul complex. My question will be about the latter notion.
Let $X$ be a smooth ...
4
votes
1
answer
309
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Compatibility between the functors of $\mathcal{O}_X$-modules and $\mathcal{D}_X$-modules
Let $f:X\to Y$ be a morphism between smooth algebraic varieties over $\mathbb{C}$.
We have natural functors $f^!:\mathsf{D}_{\text{qc}}(\mathcal{D}_Y)\to \mathsf{D}_{\text{qc}}(\mathcal{D}_X)$, $f_*:\...
7
votes
0
answers
193
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A reference for Bernstein's approach to KL conjectures
The exposition in the famous J. Bernstein's lecture notes on D-modules is rather sketchy.
Fortunately, there are many other sources for the information from this text (like Hotta's excellent "D-...
3
votes
0
answers
256
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An algebraic proof: A line bundle on a curve with a connection must be of degree 0
Let me state it using the language of $D$-modules. Let $X$ be a smooth projective curve and $\mathcal{L}$ a line bundle on it. Assume that $\mathcal{L}$ has a left action of $\mathcal{D}_X$. Then show ...
4
votes
1
answer
387
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Induced D-modules
According to the usual definition, an induced D-module on a complex manifold $X$ is a right D-module of the form $L \otimes_{\mathscr{O}_X} \mathscr{D}_X$, for $L$ a coherent sheaf of $\mathscr{O}_X$-...
3
votes
1
answer
232
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do hyperfunction solutions always exist?
I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...